IEEE TRANSACTIONS ON INDUSTRY AND GENERAL APPLICATIONS, VOL. IGA-6, NO. 1, JANUARY/FEBRUARY 1970

Design Considerations for Microwave Oven CavitiesHAROLD S. HAUCK, MEMBER, IEEE

Abstract-The heating mechanism in a microwave oven resultsfrom the interaction of a high-frequency electromagnetic field andthe food within it. In most ovens the field is generated by a magne-tron which converts dc energy to high-frequency energy at approx-imately 2450 MHz. This energy is then propagated into the ovenchamber which is, in reality, a multimode resonant cavity where theenergy is reflected from the walls to create standing wave patterns.To properly design a microwave oven, one must consider not onlythe magnetron but also the dimensions and resonant properties ofthe oven cavity. The designer must strive for a uniform energydensity within the cavity for uniform cooking of the food and mustbe sure the cavity presents the proper load to the magnetron. Theseobjectives are discussed, and a procedure for obtaining them isoutlined.

MODES WITHIN AN OVEN CAVITY

JF WE HAVE a waveguide with the wave propagating'along the Z axis as shown in Fig. 1, and then constructtwo shorting plates as shown, we have created a micro-wave cavity. This cavity is resonant at a particular setof frequencies. For instance, if the E field varies as shownin Fig. 2, where we have two half-waves in the widthdirection, three half-waves in the depth direction, andtwo half-waves in the height direction, the cavity wouldbe resonating in the TE232 mode.The wavelength at which the resonance can occur is

given by

2

\VI(MIW)2 + (N/D)2 + (P/H)2

where X is the wavelength of the resonant frequency;M, N, and P are the wavenumbers or half-waves ofvariation of E field in the W, D, and H directions, respec-tively; and W, D, and H are the width, depth, and heightof the cavity.

Since energy in a waveguide can propagate in both TEand TM modes, we can have TE and TM resonantmodes in a cavity. Thus we will also have a TM232 mode,given by the same equation. Since both modes can occurat the same frequency but have different field configura-tions, they are called degenerate modes. All modes forwhich the equation holds true can be excited in the cavityexcept for configurations which violate boundary condi-tions. These are the TEMNO modes, the TEoop modes, theTMONP modes, and the TMMop modes. Since waveguideswill not propagate the TEM mode, this mode is alsoimpossible to sustain in a cavity of this geometry.

Paper 69 TP 64-IGA, approved by the Domestic ApplianceCommittee of the IEEE IGA Group for presentation at the IEEE20th Annual Appliance Technical Conference, Detroit, Mich.,April 29-30, 1969. Manuscript received August 28, 1969.The author is with the Amperex Electronic Corporation, Hicks-

ville, N. Y. 11802.

z DIRECTION OFENERGY PROPAGATION

71IT-

I

Ll-

IIII

III

MICROWAVE CAVITYb--

Fig. 1. Waveguide with two shorting platesto form microwave cavity.

z

TE232 MODE

HEIGHT

X1IHy J 'Y

Fig. 2. Microwave cavity formed in Fig. 1 with schematic portrayalof E-field variation for the TE232 mode.

Because of the field variations due to these modes,areas of high and low energy density will occur in thecavity. If food were placed in such a cavity, it is possiblethat it may not be cooked uniformly. We have foundthat cooking uniformly can be increased by, first, choosingdimensions which give the largest number of resonantmodes within the frequency limits of the magnetron (2425-2475 MHz); and second, adding a field stirrer to constantlychange the energy pattern within the oven.

CALCULATION OF MODESA computer program can be written to compute cavity

dimensions which have a large number of resonant modes.Essentially, the equation is solved for all combinatiornsof M, N, and P for small increments of the dimensions,and the combinations which result in a resonant frequencywithin the desired frequency limits are retained. Twonomographs are also available to facilitate the calculationof modes and cavity dimensions. These nomographs areshown in Figs. 3 [1] and 4 [2] and are described in detailin Appendices I and II. The first nomograph is better for

74

HAUCK: DESIGN MICROWAVE OVEN CAVITIES

(W2 WiDTH 2H J HEIGHT)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

p=89 FREQUENCY: 2425-2475 MHz

P = 6 WAVE LENGTH: 12.12-12.37 cm.P=5

P= 4

P2

P= 117 1a76 15 14 13 .70 12 , 11 10 60 19 i8 i7 ,50 i6 i5 4o04 i3 .3o2 2011 00

041 * e61 0 *A * *21 * *01

042 * *22 * *02 (w)2 (WIDTH)2*82

64*6 *0

* 567 * **0 *68 * .68* .0.08.69 * 4*49. S09

0.5 1.0

S *43 0 *23 *g03

.44 a 924* 04 -

0 * .005 (HEIGHT)

1.5 2.0 2.I

HL

!.5

(W) = WIDTH)2

Fig. 3. Nomograph for resonant modes of rectangular microwave oven.

TM AND TE3N TM AND TE4N TM AND TE5NMODES MODES MODES

V\/ 46 V28 37 38 47

TM AND TM AND TM ANDTE6, TE7N TE8NMODES MODES MODES

56 57 66 \75

TE80 TE90

(2 DEPTH)

Fig. 4. Mode contours for rectangular cavity; TEMNp and TMMNP modes.

75

.84

68

em8at *

(w)2

3.0 3.5~~~~~~~~~~~~~~~~~~~~~~ N3 7 898

f t N A _ t ~~~~~~~~~~~~~~~~~~~6

TEO9

TE08

TE07

TE06

(2 WIDTH)

TE05

TE04

TE03

TM AND TEM8MODES

TM AND TEM7MODES

TM AND TEM6MODES

TM AND TEMOMODES

TM AND TEM4MODES

TM AND TEM3MODES

S63

TE30 TE40 TE50 TE60 TE70

IEEE TRANSACTIONS ON INDUSTRY AND GENERAL APPLICATIONS, JANUARY/FEBRUARY 1970

LEVELER VERT. HOR.IVERT. +

L

Fig. 5. Sweep frequency test circuit.

zLu

p

z

-j

a:

w

FREQUENCY - MHz

Fig. 6. Voltage reflection coefficient versu s frequency showingresonant modes in oven cavity with width 29.7 cm, depth 36.6cm, and height 30.2 cm.

calculating modes in a cavity of known dimensions, whilethe second nomograph is better for calculating dimensionswhich will have a large number of modes.To verify the existence of these modes, we calculated the

modes for a cavity with width of 29.7 cm, depth of 36.6cm, and height of 30.2 cm.

Mode

431243342333060252

Frequency(MHz)

241524302441245324622488

The cavity was then tested using the circuit of Fig. 5.In the system we used the sweep oscillator to vary the

frequency from 2400 to 2500 MHz; a dual directionalcoupler to sample the energy coming from the oscillatorand the energy being reflected from the cavity; andthe ratiometer to convert the two signals to the amplitudeof the voltage reflection coefficient. This is a measure

of the ratio of reflected to incident energy. The outpuTfrom the ratiometer can be fed into either an oscilloscopeor an X-Y recorder.

Fig. 6 shows the results using an X-Y recorder. At aresonant frequency, energy will be absorbed by the cavityand will not be reflected. At a frequency which is notresonant, the energy will be reflected. As seen in thefigure, we have resonances occurring at approximatelythe calculated frequencies for the 431, 243, 342, and252 modes. We do not see the 333 mode, and we have aslight trace of the 060 mode. We do have evidence of2 additional modes near 2500 M\IHz. Possibly these aredue to resonances within the coupling system, or thevmight be the appearance of higher frequeincy modescaused by our errors in measuring the cavity. The figureshows that most, but not necessarily all, of the calculatedmodes will appear in the cavity. Whether or not a modeappears is largely determined by the manner in which theenergy is coupled into the cavity.

FIELD STIRRER

The mode pattern within the cavity can be continuallydisturbed by adding a field stirrer, which is a bladeddevice similar to a fan and which rotates at a frequencywhich should be asynchronous with the 60-Hz powerfrequency. The effect of a stirrer can be seen in Fig. 7,which shows the heating pattern for an oven cavity withand without a stirrer. The patterns were obtained byplacing a layer of moistened silica gel in the cavity andoperating the miagnetron to see how uniformly the silicagel dries. Silica gel is normally used as a desicant and islight pink when moist but turns various shadesaif blue asit becomes dry. The light areas in the figure show areaswhich were not being heated by the energy within theoven. As seen in the figure, the uniformity of heating wasimproved by the field stirrer.As a demonstration of what is actually happening in

the cavity as the stirrer rotates, we took pictures ofthe heating pattern for six consecutive 2-degree incre-ments of rotation of the stirrer. These are shown in Fig. 8.A close examination of the figure will show that theareas of high and low energy density appear and disappear,and that, at the same time, the areas appear to movearound slightly within the oven. A further demonstrationof what is happening is given in Fig. 9, showing X-Yrecorder traces of voltage reflection coefficient versusfrequency for the 2-degree increments of stirrer rotation.The figure shows the modes appearing and disappearing,and at the same time, changing frequency slightly. Thechanges in frequency suggest that the stirrer changes theeffective dimensions of the cavity as it rotates.

LOAD PRESENTED TO THE MAGNETRON

Fig. 10 shows a magnetron operating into a transmissionline which is terminated in a load. If the load has the sameimpedance as the transmission line, the load is said to bematched, and will absorb all the energy propagating to it..

76

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I

IMOMM-11

HAUCK: DESIGN MICROWAVE OVIEN CAVITIES

Fig. 7. Silica gel patterns. (a) Without stirrer. (b) With stirrer.

: +6 I +8 a +W10 :O

Fig. 8. Silica gel patterns; six consecutive 2-degree tlirlis of stirrerin cavity.

Z 00-

060

540v

2400 2425 2450 2475 2500FREQUENCY -MH.

060

2400 2425 2450 2475 2500FREQUENCY - MH.

I *0

2400 2425 2450 2475 2500FREQUENCY -MHz

5100 -

40

2400 2425 2450 2475 2500FREQUENCY - MHz

Fig. 9. Voltage reflection coefficient versus frequency; four consecutive 2-degree turnsof stirrer in cavity.

If the load does not have the same impedance as theline, the existing mismatch will cause part of the energy

to be reflected, the amount of energy reflected beingdetermined by the degree of mismatch. As a manifes-tation of this reflected energy, a standing wave pattern,shown schematically in the figure, will be created on

the transmission line. This can be characterized by thevoltage standing wave ratio, which is the ratio of thevoltage at the maximum to the voltage at the minimumof the standing wave pattern, and by the position of the

minimum expressed in wavelengths from the referenceplane at the output of the magnetron. It can also becharacterized by the magnitude and phase of the voltagereflection coefficient mentioned earlier. Changes in theload at the end of the transmission line will result inchanges in the magnitude and phase of the standing wavepattern. The effect of these variations is shown in theload diagram of Fig. 11. Contours of constant power out-put and constant frequency are plotted versus magnitudeand phase of the voltage standing wave ratio. The phase is

77

IEEE TRANSACTIONS ON INDUSTRY AND GENERAL APPLICATIONS, JANUARY/FEBRUARY 1970

MAGNETRON

STANDING WAVE

TOWARD LOAD

- PHASE

REFERENCE PLANE

Fig. 10. Magnetron operated into a transmission line.

PHASE OFSINK REGION

.15X

F-245OMHzAVERAGE ANODE CURRENT =.380mA

PEAK ANODE CURRENT=I.IAMPSUNFILTERED RECTIFIED ANODE SUPPLY

Fig. 11. Load diagram (Amperex DX-206 magnetron).

given in fractions of a wavelength from the referenceplane at the output of the magnetron, and VSWR isgiven by concentric circles around a matched load at thecenter of the diagram.As the mismatch is increased in the phase of sink region,

the power-out increases. If the mismatch increases in theregion 1/4 wavelength from the phase of sink, the power-

out decreases. Changes in mismatch and phase also cause

changes of the tube frequency, given in the diagram as

deviations in MHz from the frequency at a matched load.For proper operation of the magnetron, the oven

cavity should present as near a matched load as possible.If mismatches do occur it is desirable that they be near thephase of sink region at the center frequency of the cookingband (2450 MHz).

DESIGN OF OVEN CAVITYThe oven cavity should have a good heating pattern

and should present the proper load to the magnetron.Although the method for obtaining these results is more

of an art than a science at the present time, definiteprocedures can be recommended. The first step is toobtain the proper oven dimensions. The rough range of

dimensions will probably be dictated by constraints otherthan electrical, such as the overall size of the applianceand the types of foods to be cooked. The final dimensionswithin these rough limits should give as large a distributionof resonant modes within the range of 2425 to 2475 MHz,since the larger the number of modes the larger the prob-ability of coupling these modes into the cavity. It is thennecessary to obtain a coupling system, including thelocation of the coupling into the oven cavity, and a stirrerdesign to present the proper VSWR and phase to themagnetron.To facilitate our work we use the test setup shown in

Fig. 5. With this equipment we can change the cavitydesign parameters at will and immediately see its effecton the magnitude of the voltage reflection coefficient.The coupling adaptor in Fig. 5 allows the equipment to"see" the oven cavity as the magnetron will see it. It ispossible to make single-frequency measurements using theequipment of Fig. 12, but the procedures are more timeconsuming. More sophisticated equipment showing bothmagnitude and phase of voltage reflection coefficient isalso available.A typical waveguide coupler using a 2- by 4-inch wave-

guide to transmit the energy from the magnetron to theoven cavity is shown in Fig. 13. The opening into thecavity is 2 by 4 inches, and the center of the opening forthe magnetron output is 1.171 inches from the oppositeend of the waveguide adaptor. For other size waveguidesthe distance from the tube output to the back of thewaveguide must be determined by trial. The 2- by 4-inchopening is essentially an antenna into the oven cavity.

Quite often the most convenient location for the wave-guide opening into the cavity is in the top or the back wall,depending upon how the various components must bearrainged in the appliance. We have had good results byplacing the waveguide opening near one of the corners ofthe back wall. We have also had good results by position-ing the stirrer so that the blades pass in front of theopening. Uniformity of cooking pattern is usually in-creased if each blade of the stirrer is at a different pitch,for instance, 10, 20, 30, and 40 degrees from the plane ofthe hub for a four-bladed stirrer. In general, we havefound that the position of the back plate of the wave-guide feed and the position and orientation of the openingaffect, mostly, the magnitude of the VSWR; the lengthof the waveguide feed affects the phase; and the stirrerdesign affects the uniformity of cooking.When the position and orientation of the antenna

opening and the stirrer design have been adjusted togive the best matched load over the frequency range, it isthen necessary to measure the phase and VSWR at singlefrequencies, say 2425, 2450, and 2475 MHz. We normallymeasure at 16 uniformly spaced positions of the stirrerblade at each frequency. From these measurements onecan tell whether the length of the waveguide coupler mustbe adjusted to put the phase of the load, represented bythe cavity, in the best location with respect to the mag-netron reference plane.

78

HAUCK: DESIGN MICROWAVE OVEN CAVITIES

-11111a:_ LEVELER VERT.-I _ ~ _. HOR..+ I

-I

Fig. 12. Single-frequency test circuit.

Fig. 13. Typical waveguide coupler used to direct energyfrom Amperex DX-206 magnetron to oven cavity.

When the oven cavity has been properly adjusted usingthe test instrumentation, the magnetron can then beattached and operated to determine the uniformity ofheating. This can be done using the silica gel techniquedescribed earlier and by spacing various foods at regularintervals throughout the oven. A good test of patternuniformity is to bake a cake in the oven. With a poorpattern, the cake will rise unevenly. If the pattern is notsufficiently uniform, adjustments in the stirrer andpossibly the opening position and orientation must bemade. This being the case, the oven will then have to bereevaluated to be sure it still presents the proper load tothe magnetron. When the VSWR, phase, and pattern aresatisfactory, the design of the cavity and coupling arecomplete. The objectives of good heating uniformity andproper load for the magnetron will have been obtained.

CONCLUSIONCareful design of the oven cavity following the pro-

cedures outlined will give optimum operation of themagnetron and will lead to more efficient cooking. Thetechniques which have been given closely follow theoreticalconsiderations and give excellent results.

APPENDIX I

The nomograph of Fig. 3 can be used as indicated by theexample in the lower right corner of the figure.

1) The ratios (W/H)2 = (width/height)2, (W/X)2(width/wavelength) 2, and (W/D)2 = (width/depth)2must be calculated using both wavelengths, 12.12 and12.37 cm, corresponding to 2475 and 2425 MHz, respec-tively.

2) Draw lines from (W/H)2 on the upper scale to bothvalues of (W/X)2 on the center scale corresponding to theP = 0 line.

3) Draw lines from both values of (W/X)2 on the centerscale to the value of (W/D)2 on the bottom scale. Alldots in the lower portion of the nomograph occurringwithin this wedge correspond to modes with P = 0occurring within the wavelength limits 12.12 to 12.37cm.

4) Draw lines from the origin of the (W/H)2 axisthrough the intersection of the lines of step 2) with theP = 1 line to the center scale. Connect these newlyobtained points on the center scale with (W/D)2 on thelower scale. All crosses within the new wedge correspondto modes with P = 1.

5) Repeat step 4) using the P = 2 line to obtain modeswith P = 2. This step is then repeated for all P linescrossed by the lines of step 2).

APPENDIX II

In [2j Griemsmann gave a graph for determining modesin an oversized waveguide. Using the equation

JJ/2 N2 1 1

(2W/X)2 (2D/X)2 sec20 csc26(1)

where W and D are the transverse dimensions of the wave-guide and M, N, and X are given in Fig. 2, he obtainedthe parametric equations

W M sec 0 and - = N csc 0.x x (2)

These were used to obtain the plot of 2D/X versus 2W/Xfor all TE and TM modes with mode numbers up to5, covering the range 1 < 2D/X < 10 and 1 < 2W/X <10. The plotting points for the graph were obtained bychoosing values of M, N, and 6 to satisfy the equations

1 < MsecO < 10 and 1 < Ncsc < 10. (3)

For our work we constructed a similar but largergraph covering the range of 2W/X and 2D/X from 3 to 10,and including modes with mode numbers up to 8. Fig. 4is the resulting graph. Each curve represents a mode orpair of 4egenerate modes as indicated around the pe-riphery. ^To use the graph for calculations of rectangular cavities,

(1) can be modified as follows. Consider the equation forrectangular cavities:

M2 N2 p2(2W/) + ( ) + -/= 1.(2W/X)2 (2D/X)2 (2H/X)2 (4)

This can be rewritten as

M2 N2 p2+ N = 1- P - F2(2W/X)2 (2D/X)2 (2H/X)2

79

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IEEE TRANSACTIONS ON INDUSTRY AND GENERAL APPLICATIONS, JANUARY/FEBRUARY 1970

or

M2 N2= 1 (5)(F 2W/X) 2 ' (F.2D/X)2

where

F= (21/X)2= tanh Lcosh-1(2)H (6)

To calculate the modes in a cavity, we must thuscalculate the factor F from (6) from the height H of thecavity and the mode number P. F can then be substitutedinto (5) to obtain the quantities F. 2W/X and F. 2D/X. Toenter the graph, the graph can be used with either set ofcoordinates 2W/X 2D/X or (F. 2W/X) (F. 2D/X). As inthe case of Fig. 2, the modes which fall within the wave-length range of 12.12-12.37 cm are the cavity modes ofinterest.

Fig. 4 can be used to calculate the dimensions which willgive a large number of modes by looking for clusters ofseveral mode lines and calculating the dimensions fromthe coordinates of these clusters. The width and depthare obtained from the coordinates 2W/X and 2D/X.The height is then calculated from F- 2W/X and F. 2D/Xusing (6). One may use the same cluster of modes tocalculate the height, in which case F = 1, or one canlook for another cluster of modes with the coordinatesF. 2W/X and F.2D/X. In either case, values of P aresubstituted into (6) until a convenient height is obtained.

REFERENCES[1] R. N. Bracewell, "Charts for resonant frequencies of cavities,"

Proc. IRE (Waves and Electrons), vol. 35, pp. 830-841, August1947.

[2] J. W. E. Griemsmann, "Oversized waveguides," Microwaves,vol. 2, pp. 20-31, December 1963. (The method for using thegraph for rectangular cavities was given by Dr. Griemsmann inprivate correspondence.)

Harold S. Hauck 4'56) was born on November 3, 1931. He received the A.B. degree inmathematics and the M.S. degree in engineering from the University of Pennsylvania,Philadelphia, in 1954 and 1964, respectively. He is presently working toward the M.S. degreein industrial management at the Polytechnic Institute of Brooklyn, Brooklyn, N. Y.Upon graduation in 1954 he joined the Research Laboratory of the Polymer Corporation,

where he was responsible for research into the dielectric, dynamic mechanical, and frictionalproperties of plastics. While there, he developed a filled-plastic magnetic material for use inRF, UHF, and microwave circuits. In 1961 he joined the Plastics Engineering Laboratory ofRohm and Haas Company. His principle duties were to study the plate and shell properties ofstructures made from acrylic plastics for architectural applications. Since 1966 he has beenemployed by the Amperex Electronic Corporation, Hicksville, N. Y., in the areas of magnetrondevelopment and applications research and assistance in microwave oven design.

so